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\noindent Name: \rule{7cm}{.2pt} \vspace{.3cm} \hfill Date:  \rule{3.5cm}{.2pt}
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\begin{center}
{\bf \Large{MA 162}} \\\vspace{.1cm} Week 14 Recitation Worksheet (Thursday)
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\fbox{\textbf{You must show all work to receive full credit.}}
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\noindent With both simple interest and compound interest, the amount of money deposited was a lump sum and was left to accumulate interest.  We will now consider a few examples in which \emph{payments are made on a regular basis into an account}.  An \BLUE{annuity} is a sequence of equal payments made at equal intervals of time.  We will only consider \emph{ordinary annuities}, meaning that the payments are made at the end of each period.  We will also assume the frequency of payments is the same as the frequency of compounding interest.  If a payment $R$ is deposited at the end of each compounding period into an account earning an annual interest rate $r$ for $t$ years, then the accumulated value (or future value) $A$ of the annuity is 
\begin{equation*}
A = R\left[\frac{(1+\frac{r}{n})^{nt}-1}{\frac{r}{n}}\right],
\end{equation*}
where $n$ is the number of compounding periods per year.\vspace{0.5cm}

\begin{enumerate}[ref=\#\arabic*]
%1
\item (HW19 \#1) If you make quarterly payments of \$454.00 into an ordinary annuity earning an annual interest rate of 6.92\%, how much will be in the account after 5 years?\vfill


%2
\item (HW19 \#2) If you make monthly payments of \$497.00 into an ordinary annuity earning an annual interest rate of 4.79\% compounded monthly, how much will you have in the account after 3 years?\vfill


\newpage
%3
\item[3$^*$.] (HW19 \#3) If you make quarterly deposits of \$378.00 into an ordinary annuity earning an annual interest rate of 4.35\%, 
\begin{parts}
\part how much will be in the account after 6 years?\vfill
\part How much interest did you earn in those 6 years?\vspace{0.5cm}\vfill
\end{parts}


\setcounter{enumi}{3}
%4
\item (HW19 \#4) How much do you need to invest semiannually into an ordinary annuity earning an annual interest rate of 6.32\% compounded semiannually so that you will have \$6,106.91 after 8 years?\vfill


\newpage
%5
\item (HW19 \#5) In 5 years Harry and Sally would like to have \$26,000.00 for a down payment on a house.  How much should they deposit each month into an account paying 14\% compounded monthly?\vfill


%6
\item[6$^*$.] (HW19 \#6) Irene plans to retire on December 31st, 2019.  She has been preparing to retire by making annual deposits, starting on December 31st, 1979, of \$2450 into an account that pays an effective rate of interest of 9\%.  She has continued this practice every year through December 31st, 2000.  Her goal is to have \$1.45 million saved up at the time of her retirement.  How large should her annual deposits be (from December 31st, 2001 until December 31st, 2019) so that she can reach her goal?\vfill
\end{enumerate}


\newpage
\noindent \BLUE{Amortization} is a process by which one pays down a loan or withdraws funds from an account in equal installments.\\\\
If a periodic payment $R$ is made towards a loan (or received from an account) earning an annual interest rate $r$ for $t$ years, then the principal of the loan $P$ (or principal of the account) is
\begin{equation*}
P=R\left[\dfrac{1-\left(1+\frac{r}{n}\right)^{-nt}}{\frac{r}{n}}\right],
\end{equation*}
where $n$ is the number of compounding periods per year (also the number of payments per year).\vspace{0.5cm}

\begin{enumerate}[ref=\#\arabic*]
\setcounter{enumi}{6}
\item[7$^*$.] (HW20 \#2) A couple has decided to purchase a \$100000 house using a down payment of \$19000. They can amortize the balance at 10\% over 15 years. 
\begin{parts}
\part What is their monthly payment?\vfill
\part What is the total interest paid?\vfill
\part What is the equity after 5 years?\vfill
\end{parts}


\newpage
\setcounter{enumi}{7}
\item (HW20 \#3) Fifteen years ago a couple purchased a house for \$150,000.00 by paying a 20\% down payment and financing the remaining balance with a 30-year mortgage at 4.62\% compounded monthly. 
\begin{parts}
\part Find the monthly payment for this loan.\vfill\vfill
\part Find the balance of the loan after 19 years and after 20 years.  Let $n$ be how many payments are left on the loan.
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\hline\bigstrut
After 19 years & \hspace{2.25cm} & After 20 years & \hspace{2.25cm} \\
\hline
$n = $ & \hspace{2.25cm} & $n = $ & \hspace{2.25cm} \bigstrut\\
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Loan Balance:  \$ & \hspace{2.25cm} & Loan Balance:  \$ & \hspace{2.25cm} \bigstrut\\
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\end{center}\vfill\vfill\vfill
\part Find the total amount of interest paid by the couple during the 20th year.\vfill
\end{parts}


\newpage
\item (HW20 \#5) Juan purchases an annuity for \$4240 that will make 23 annual payments, the first to come in one year.   If the effective rate of interest is 8.8\%, how much is each annual payment?\vfill
\end{enumerate}
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